Compositional data are common in many fields, both as outcomes and predictor variables. The inventory of models for the case when both the outcome and predictor variables are compositional is limited, and the existing models are often difficult to interpret in the compositional space, due to their use of complex log-ratio transformations. We develop a transformation-free linear regression model where the expected value of the compositional outcome is expressed as a single Markov transition from the compositional predictor. Our approach is based on estimating equations thereby not requiring complete specification of data likelihood and is robust to different data-generating mechanisms. Our model is simple to interpret, allows for 0s and 1s in both the compositional outcome and covariates, and subsumes several interesting subcases of interest. We also develop permutation tests for linear independence and equality of effect sizes of two components of the predictor. Finally, we show that despite its simplicity, our model accurately captures the relationship between compositional data using two datasets from education and medical research.

中文翻译：

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成分结果和预测因子的无变换线性回归

成分数据在许多领域都很常见，既可以作为结果变量，也可以作为预测变量。结果和预测变量都是组合变量的情况下的模型清单是有限的，并且现有模型通常难以在组合空间中解释，因为它们使用了复杂的对数比变换。我们开发了一个无变换的线性回归模型，其中成分结果的预期值表示为来自成分预测变量的单个马尔可夫转换。我们的方法基于估计方程，因此不需要完整的数据可能性规范，并且对不同的数据生成机制具有鲁棒性。我们的模型易于解释，允许在组合结果和协变量中使用 0 和 1，并包含几个有趣的子案例。我们还开发了线性独立性和预测变量两个分量的效应大小相等的排列检验。最后，我们表明，尽管它很简单，但我们的模型使用来自教育和医学研究的两个数据集准确地捕获了成分数据之间的关系。